Level Zero Fundamental Representations over Quantized Affne Algebras and Demazure Modules

Abstract

Let W(ω\_k\_) be the finite-dimensional irreducible module over a quantized affne algebra U'q(g) with the fundamental weight ω\_k\_ as an extremal weight. We show that its crystal B(W(ω\_k\_)) is isomorphic to the Demazure crystal B\_−(−Λ0 + ω\_k). This is derived from the following general result: for a dominant integral weight λ and an integral weight µ, there exists a unique homomorphism U'q(g)(\_u\_λ ⊗ \_u\_µ) → V(λ + µ) that sends \_u\_λ ⊗ \_u\_µ to \_u\_λ+µ. Here V(λ) is the extremal weight module with λ as an extremal weight, and \_u\_λ ∈ V(λ) is the extremal weight vector of weight λ.